<!DOCTYPE html>
<html lang="en-US">
<!--********************************************-->
<!--*       Generated from PreTeXt source      *-->
<!--*                                          *-->
<!--*         https://pretextbook.org          *-->
<!--*                                          *-->
<!--********************************************-->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<meta name="robots" content="noindex, nofollow">
</head>
<body class="ignore-math">
<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Linearity</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\cal L}[a f(t)+b g(t)]=aF(s)+bG(s).
\end{equation*}
</div>
<p class="continuation">Examples:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}[3t+4] = 3\frac{1}{s^2}+4\frac{1}{s}.
\end{equation*}
</div>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}[\cos at + i\sin at]={\mathcal L}[e^{iat}] \  \ {\rm by \ DeMoivre}.
\end{equation*}
</div>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}[e^{iat}] = \frac{1}{s-ia}=\frac{s}{s^2+a^2} + \frac{ia}{s^2+a^2}.
\end{equation*}
</div>
<p class="continuation">Hence, equating real and imaginary parts and using linearity</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}[\cos at]=\frac{s}{s^2+a^2}
\end{equation*}
</div>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}[\sin at]=\frac{a}{s^2+a^2}.
\end{equation*}
</div>
<span class="incontext"><a href="sec8_2.html#p-440" class="internal">in-context</a></span>
</body>
</html>
